Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 39-49.

A boundary control problem with a nonlinear reaction term

John R. Cannon, Mohamed Salman

The authors study the problem $u_t=u_{xx}-au$, $0<x<1$, $t>0$; $u(x,0)=0$, and $-u_x(0,t)=u_x(1,t)=\phi(t)$, where $a=a(x,t,u)$, and $\phi(t)=1$ for $t_{2k} < t<t_{2k+1}$ and $\phi(t)=0$ for $t_{2k+1} <t<t_{2k+2}$, $k=0,1,2,\ldots$ with $t_0=0$ and the sequence $t_{k}$ is determined by the equations $\int_0^1 u(x,t_k)dx = M$, for $k=1,3,5,\dots$, and $\int_0^1 u(x,t_k)dx = m$, for $k=2,4,6,\dots$, where $0<m<M$. Note that the switching points $t_k$, are unknown. A maximum principal argument has been used to prove that the solution is positive under certain conditions. Existence and uniqueness are demonstrated. Theoretical estimates of the $t_k$ and $t_{k+1}-t_k$ are obtained and numerical verifications of the estimates are presented.

Published April 15, 2009.
Math Subject Classifications: 35K57, 35K55.
Key Words: Reaction-diffusion, Parabolic, Nonlocal boundary conditions.

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John R. Cannon
University of Central Florida, Department of Mathematics
Orlando, FL 32816, USA
Mohamed Salman
Tuskegee University, Department of Mathematics
Tuskegee, AL 36088, USA

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